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The Combined Effects of Gravitational and Thermocapillary Driving Forces on the Interactions of Slightly Deformable, Surfactant - Free Drops

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Abstract

By an asymptotic approach previously employed for gravitational or thermocapillary motion alone, collision efficiencies are calculated for slightly deformable drops in combined gravitational and thermocapillary motion with negligible inertia and thermal convection. The constant imposed temperature gradient may be aligned with gravity in either the same or opposite direction. In the dimensionless parameter space, deformation becomes important at a smaller drop size ratio when the temperature gradient and gravity are aligned in the same direction, because the driving force is larger and induces dimple formation earlier. For the same reason, in a physical system of ethyl salicylate (ES) drops in an unbounded matrix of diethylene glycol (DEG), deformation becomes important for smaller drops when the driving forces have a parallel, rather than anti-parallel, arrangement. In developing the population dynamics for slightly deformable drops, a new, simplified expression for the collision efficiency for spherical drops in the absence of van der Waals forces is presented, which successfully separates the contributions of the two driving forces. Two collision-forbidden regions can occur for opposed driving forces leading to a shark-fin shaped collision efficiency curve for two slightly deformable drops. As shown in population dynamics, if the drop distribution is broad enough, it is possible for drops to jump the first collision-forbidden region.

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Abbreviations

A :

Hamaker constant, erg

a〉:

volume-averaged drop radius, cm

a i :

ith drop radius, μm

b :

scaling value of dimple radius, μm

C a :

capillary number, \(\mu _{e} V_{G,12}^{(0)}/\gamma _{0}\)

D T :

thermal diffusivity, cm2/s

\(d_{\infty }\) :

initial horizontal offset at infinite vertical separation, cm

E :

collision efficiency, [yc/(a1 + a2)]2

f :

drop distribution function

f t :

dimensionless tangential stress

G :

parallel mobility function for equal but opposite external forces

g :

gravitational constant, m/s2

h :

dimensionless drop gap, R(ra1a2)/b2

J 12 :

collision rate per unit volume, (cm3s)− 1

k :

drop-size ratio, a1/a2

k i :

dispersed or external phase thermal conductivity, W/mK

\(\hat {k}\) :

thermal conductivity ratio, kd/ke

L :

parallel gravitational mobility function

L M :

parallel thermocapillary mobility function

M :

transverse gravitational mobility function

M M :

transverse thermocapillary mobility function

N F :

modified velocity ratio

N V :

velocity ratio, \(\pm V_{G,12}^{(0)}/V_{M,12}^{(0)}\)

n :

number of drops per unit volume, cm− 3

p :

dimensionless pressure

p 12 :

pair distribution function

Q 12 :

interparticle force parameter

R :

reduced drop radius, cm

R e :

Reynolds number, \(\rho _{e} V_{G,12}^{(0)} a_{2}/\mu _{e}\)

r :

center-to-center drop distance, cm

s :

dimensionless center-to-center drop distance, 2r/(a1 + a2)

T :

temperature, oC or K

t :

dimensionless time

t i :

time-scale, s

\(V_{12}^{(0)}\) :

relative drop velocity at infinite separation, cm/s

y c :

critical horizontal offset at infinite vertical separation, cm

α :

dimensionless contact force

β :

angle between vertical and the drops’ line of centers, rad

γ :

interfacial tension, dyn/cm

δ :

dimensionless Hamaker parameter

ζ :

dimensionless angular coefficient

μ i :

dispersed or external phase viscosity, g/cms

\(\hat {\mu }\) :

drop-to-medium viscosity ratio, μd/μe

ξ :

dimensionless gap between drops, s − 2

ρ i :

dispersed or external phase density, g/cm3

\(\hat \sigma \) :

dimensionless standard deviation

ϕ :

Green’s function for axisymmetric flow

ϕ 0 :

volume fraction of dispersed phase

ω :

dimensionless parameter, αζ

0:

initial or reference value

1:

smaller drop

2:

larger drop

cr :

critical

d :

dispersed or drop phase

e :

external or matrix phase

G :

gravitational

M :

Marangoni-induced or thermocapillary

C :

combined gravitational and thermocapillary

G :

gravitational

T :

thermocapillary

\({\hat {~~}}\) :

dimensionless or modified

\(\nabla T_{\infty }\) :

applied temperature gradient, K/cm

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Acknowledgments

The authors would like to thank the Minnesota Supercomputing Institute for the use of computing resources.

Author information

Correspondence to Michael A. Rother.

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Appendices

Appendix A

The governing equations for the inner region where two drops are nearly touching, namely the thin-film equations, are presented here in dimensionless form (Rother et al. 1997).

Normal Stress Balance

$$ p-\frac{\delta}{h^{3}}=2-\frac{1}{r} \frac{\partial}{\partial r} \left( r\frac{\partial h}{\partial r}\right) $$
(A.1)

Momentum Balance

$$ f_{t}=-\frac{h}{2} \frac{\partial p}{\partial r} $$
(A.2)

Local Boundary Integral

$$ f_{t}\left( r\right)=4{\int}_{0}^{\infty} \phi\left( r^{\prime},r\right) \left[\frac{u}{r^{\prime2}} - \frac{1}{r^{\prime}} \frac{\partial u}{\partial r^{\prime}} - \frac{\partial^{2}u}{\partial r^{\prime2}}\right]dr^{\prime} $$
(A.3)

Mass Continuity

$$ \frac{\partial h}{\partial t} + \frac{1}{r} \frac{\partial} {\partial r}\left( rhu\right)=0 $$
(A.4)

Integral Force Balance

$$ {\int}_{0}^{\infty} \left( p-\frac{\delta}{h^{3}}\right)rdr= \left[\alpha_{G} \pm N_{F} \alpha_{M}\right]\cos\beta\left( t\right) $$
(A.5)

Appendix B

The derivation for Eq. 13, a simplified expression for the collision efficiency \({E_{0}^{C}}\) of two spherical drops in combined gravitational and thermocapillary motion without van der Waals forces, is provided here. In the absence of deformation and attractive molecular forces, the governing equations for the critical horizontal \(y_{cr}^{C}\) for two drops interacting in the presence of gravity and a vertical temperature gradient with negligible inertia are linear, and the flow is reversible. Moreover, the velocity of the drops is linearly related to the sum of the driving forces and depends on the instantaneous position. We note here that the critical horizontal offset ycr corresponds to the critical value of \(d_{\infty }\) shown in Fig. 3 demarcating the boundary between trajectories leading to collision and separation of the two drops.

In this work we consider dilute dispersions of drops such that the probability of a three-body collision is small and seek to predict the average collision rate between drops of size category 1 with those of size category 2 at a given time. As discussed elsewhere (Zhang and Davis 1991; Manga and Stone 1995), the collision rate J12 per unit volume, which is equal to the flux of drop pairs into the collision surface r = a1 + a2, is given by

$$ J_{12} = -n_{1} n_{2} {\int}_{r = a_{1}+a_{2}} p_{12} \mathbf{V}_{\mathbf{12}} \cdot \mathbf{n} dA, $$
(B.1)

where p12(r) is the pair-distribution function, V12 is the drop relative velocity, n = r/r is the outward pointing normal vector to the spherical surface r = a1 + a2, and ni is the number of drops per unit volume in size categories i = 1 or 2, characterized by radius a1 and a2, respectively.

In the case of a dilute dispersion, the pair-distribution function must satisfy a quasi-steady mass conservation equation outside the contact surface (Zhang and Davis 1991; Manga and Stone 1995):

$$ \nabla \cdot \left( p_{12} \mathbf{V}_{\mathbf{1}\mathbf{2}} \right) = 0. $$
(B.2)

Because the drops coalesce as they come into contact, p12 = 0 at r = a1 + a2. In addition, because the drop interactions begin at wide separations, the other boundary condition is that p12 → 1 as \(r \to \infty \).

For the case of gravitational motion alone, thermocapillary motion alone, or combined gravitational and thermocapillary motion, the same technique can be used to find the collision rate and collision efficiency based on the symmetry of the flows. By employing (B.2) and the divergence theorem, one integrates (B.1) over the surface enclosing the volume of all trajectories starting at r = \(\infty \) and ending in contact. At infinite separation, the cross-section of this volume from symmetry considerations is a circle with radius ycr. Because p12 = 1 and V12 = \(\mathbf {V}_{\mathbf {12}}^{\mathbf {(0)}}\) at r = \(\infty \), the collision rate is

$$ J_{12,i} = n_{1} n_{2} V_{12.i}^{(0)} \pi (y_{cr}^{i})^{2}, $$
(B.3)

where i = G, T, or C for gravitational, thermocapillary or combined motion, respectively. In the Smoluchowski limit, where there are no hydrodynamic interactions, the critical offset is ycr = a1 + a2 and thus the collision efficiency for spherical drops in the absence of attractive molecular forces is

$$ {E_{0}^{i}} = \left( {y_{cr}^{i} \over a_{1} + a_{2}}\right)^{2}, $$
(B.4)

where again i = G, T or C for the appropriate motion.

Now consider the case of the collision rate for combined gravity and thermocapillarity after application of the divergence theorem:

$$ \begin{array}{@{}rcl@{}} J_{12,C} &=& n_{1} n_{2} {\int}_{r = a_{1}+a_{2}} \nabla \cdot \left( p_{12} \mathbf{V_{12,C}} \right) dV\\ &&- n_{1} n_{2} {\int}_{r = \infty} p_{12} \mathbf{V_{12,C}} \cdot \mathbf{n} dA \end{array} $$
(B.5A)
$$ \begin{array}{@{}rcl@{}} &=& n_{1} n_{2} {\int}_{r = a_{1}+a_{2}} \nabla \cdot \left( p_{12} (\mathbf{V}_{\mathbf{1}\mathbf{2},\mathbf{G}} + \mathbf{V}_{\mathbf{1}\mathbf{2},\mathbf{T}}) \right) dV\\ &&- n_{1} n_{2} {\int}_{r = \infty} p_{12} \mathbf{V_{12,G}} \cdot \mathbf{n} dA\\ &&- n_{1} n_{2} {\int}_{r = \infty} p_{12} \mathbf{V_{12,T}} \cdot \mathbf{n} dA, \end{array} $$
(B.5B)

where V12, C = V12, G + V12, T. The volume integrals on the right side of Eqs. B.5A and B.5B are identically zero because of Eq. B.2. Evaluation of the surface integrals leads to

$$ J_{12,C} = n_{1} n_{2} V_{12,C}^{(0)} \pi (y_{cr}^{C})^{2} = J_{12,G} \pm J_{12,T} $$
(B.6A)
$$ \begin{array}{@{}rcl@{}} &=& n_{1} n_{2} V_{12,G}^{(0)} \pi (y_{cr}^{G})^{2} \left( {1 + {1 \over N_{V}} \over 1 + {1 \over N_{V}}} \right)\\ &\pm& n_{1} n_{2} V_{12,T}^{(0)} \pi (y_{cr}^{T})^{2} \left( {N_{V} + 1 \over N_{V} + 1} \right), \end{array} $$
(B.6B)

where the terms involving NV in large parentheses are equivalent to multiplying by one and the sum is used for driving forces aligned in the same direction and the difference for opposed driving forces.

Simplifying Eq. B.6B leads to

$$ \begin{array}{@{}rcl@{}} J_{12,C} = n_{1} n_{2} V_{12,G}^{(0)} \left( 1 + {1 \over N_{V}} \right) \left( {\pi (y_{cr}^{G})^{2} \over 1 + {1 \over N_{V}}} \right)\\ \pm n_{1} n_{2} V_{12,T}^{(0)} (N_{V} + 1) \left( {\pi (y_{cr}^{T})^{2} \over N_{V} + 1} \right) \end{array} $$
(B.7A)
$$ \begin{array}{@{}rcl@{}} &=& n_{1} n_{2} V_{12,C}^{(0)} \left( {\pi (y_{cr}^{G})^{2} \over 1 + {1 \over N_{V}}} \right)\\ &&+ n_{1} n_{2} V_{12,C}^{(0)} \left( {{1 \over N_{V}} \pi (y_{cr}^{T})^{2} \over 1 + {1 \over N_{V}}}\right) \end{array} $$
(B.7B)
$$ = n_{1} n_{2} V_{12,C}^{(0)} \left( {\pi (y_{cr}^{G})^{2} + {1 \over N_{V}} \pi (y_{cr}^{T})^{2} \over 1 + {1 \over N_{V}}} \right), $$
(B.7C)

where \(V_{12,C}^{(0)} = V_{12,G}^{(0)} \left (1 + {1 \over N_{V}} \right ) = \pm V_{12,T}^{(0)} \left (N_{V} + 1 \right )\) and the negative sign is used if NV < 0.

Dividing by the collision rate in the Smoluchowski limit provides an expression for the combined collision efficiency:

$$ {E_{0}^{C}} = {\left( {y_{cr}^{G} \over a_{1} + a_{2}} \right)^{2} +{1 \over N_{V}} \left( {y_{cr}^{T} \over a_{1} + a_{2}} \right)^{2} \over 1 + {1 \over N_{V}}} $$
(B.8A)
$$ = \left( {y_{cr}^{C} \over a_{1} + a_{2}} \right)^{2} = {{E_{0}^{G}} + {1 \over N_{V}} {E_{0}^{T}} \over 1 + {1 \over N_{V}}}. $$
(B.8B)

Thus, Eq. 13 is recovered.

Equation B.8B is exact and does not require calculations with Eq. 2 combining mobility functions for gravitational and thermocapillary motion for each value of the relative velocity parameter NV. A similar expression can also be derived for collision efficiencies for combined gravitational and thermocapillary motion of spherical drops covered with incompressible surfactant without van der Waals forces (Rother 2010) but is not presented here.

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Stark, J.K., Rother, M.A. The Combined Effects of Gravitational and Thermocapillary Driving Forces on the Interactions of Slightly Deformable, Surfactant - Free Drops. Microgravity Sci. Technol. (2020) doi:10.1007/s12217-019-09774-y

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Keywords

  • Thermocapillary
  • Gravitational
  • Drops
  • Coalescence
  • Collision Efficiency